Numerical Experiments

1. Problem Statement

We have a geometry splitted in 9 subparts :

Geometry

image::example-thermal-block-heat-transfer.png

Given \(\mu \in (\mu_1,...\mu_P) \in \mathcal{D}^\mu\equiv [\mu^{\text{min}},\mu^{\text{max}}]^P\), evaluate (recall that \(\ell = f\))
\(s(\mu) = f(u(\mu))]\) where \(u(\mu) \in X \equiv \{ v \in H^1(\Omega), v|_{\Gamma_{\text{top}}} = 0\}\) satisfies \(a(u(\mu), v; \mu) = f(v; \mu) \quad \forall v \in X\) we have \(P = 8\) and given \(1 < \mu_r < \infty\) we set \(\mu^{\mathrm{min}} = \dfrac{1}{\sqrt{\mu_r}},\quad \mu^{\mathrm{max}} = \sqrt{\mu_r}\) such that \(\dfrac{\mu^{\mathrm{max}}}{\mu^{\mathrm{min}}}=\mu_r\).

Recall we are in the compliant case \(\ell = f\), we have \(f(v) = \displaystyle\int_{\Gamma_{0}} v\quad \forall v \in X\) and \(a(u,v;\mu) = \displaystyle\sum_{i=1}^{P} \mu_i \int_{\Omega_i} \nabla u \cdot \nabla v + 1 \int_{\Omega_{P+1}} \nabla u \cdot \nabla v \quad\forall u,\ v\ \in X\) where \(\Omega = \displaystyle\cup_{i=1}^{P+1} \Omega_i\).

The inner product is defined as follows \((u,v)_X = \displaystyle\sum_{i=1}^P \bar{\mu}_i \int_{\Omega_i}\nabla u \cdot \nabla v + 1 \int_{\Omega_{P+1}} \nabla u \cdot \nabla v\) where \(\bar{\mu}_i\) is a reference parameter. We have readily that \(a\) is

  • symmetric,

  • parametrically coercive \(0 < \dfrac{1}{\sqrt{\mu_r}} \leq \mathrm{min}(\mu_1/\bar{\mu}_1, \ldots, \mu_P/\bar{\mu}_P,1) \leq \alpha(\mu)\),

  • and continuous \(\gamma(\mu) \leq \mathrm{max}(\mu_1/\bar{\mu}_1, \ldots, \mu_P/\bar{\mu}_P,1) \leq \sqrt{\mu_r} < \infty\)

and the linear form \(f\) is bounded.

2. Affine decomposition

We obtain the affine decomposition \(a(u,v;\mu) = \displaystyle\sum_{q=1}^{P+1} \Theta^q(\mu) a^q(u,v)\) with \(\begin{aligned} \Theta^1(\mu) = \mu_1 & & a^1(u,v) = \int_{\Omega_1} \nabla u \cdot \nabla v\\ & \vdots & \\ \Theta^P(\mu) = \mu_P & & a^P(u,v) = \int_{\Omega_P} \nabla u \cdot \nabla v\\ \Theta^{P+1}(\mu) = 1 & & a^{P+1}(u,v) = \int_{\Omega_{P+1}} \nabla u \cdot \nabla v \end{aligned}\)

3. Results

  • Homogeneous parameters

Maximum parameter values

image::33-max.png

Minimum parameter values

image::33-min.png

  • Heteregeneous parameters

Random values

image::33-random.png

4. Thermal Block \(P=1\)

Random values

image::thermalblock-P1.png

We assume \(1/\mu^{\min}_1=\mu^{\max}_1=\sqrt{\mu_r}=10\) and choose \(\mathcal{N}=256\)

we set \(\bar{\mu}=1\) and have \(\Theta^1_a(\mu)=\mu_1,\, \Theta^2_a(\mu) = 1\). Thus, \(\Theta_a^{\min,\bar{\mu}}(\mu_1)=\min(\mu_1,1)\\ \Theta_a^{\max,\bar{\mu}}(\mu_1)=\max(\mu_1,1)\)

hence \(\theta^{\bar{\mu}}(\mu_1) = \max(\frac{1}{\mu_1},\mu_1)\) and \(\theta^{\bar{\mu}}(\mu_1) \leq\sqrt{\mu_r} \forall \mu_1 \in \mathcal{D}\).

Values in [RHP2008]

Table 1. Convergence results for \(P=1\)
\(N\) \(\Delta^s_{N,\mathrm{max}}(\mu)\) \(\eta^s_{N,\mathrm{ave}}\) \(\eta^s_{N,\mathrm{max}}\)

1

7.2084E+00

2.3417

3.3305

2

4.5371E–01

2.4858

3.6850

3

6.9652E–04

6.2195

9.8551

4

1.3744E–07

3.3219

7.2632

5

3.1140E–11

6.0789

7.0453

Note that: \(\eta^s_{N,\mathrm{max}}(\mu_1) \leq \eta^s_{\mathrm{max,UB}} \equiv \sqrt{\mu_r} = 10\)

  • Maximum output error bound: \(\Delta^s_{N,\mathrm{max}} = \max_{\mu \in \Xi_{\mathrm{train}}} \Delta^s_N(\mu)\)

  • Average output effectivity: \(\eta^s_{N,\mathrm{ave}} = \frac{1}{\Xi_\mathrm{train}}\sum_{\mu \in \Xi_{\mathrm{train}}} \eta^s_N(\mu)\)

  • Maximum output effectivity: \(\eta^s_{N,\mathrm{max}} = \max_{\mu \in \Xi_{\mathrm{train}}} \eta^s_N(\mu)\)

5. Example Thermal Block \(P=8\)

  • Configuration :

    • 47 600 dofs;

    • Preconditionner : LU – Solver : MUMPS

    • \(\Xi : \) parameter sampling of dimension 1 000.

  • Plot \(\max_{\mu \in \Xi} \frac{ |s^{\mathcal{N}}(\mu)-s_N(\mu)|}{s^{\mathcal{N}}(\mu)}\)

image::Relative-error.png

Thermal Block \(P=8\)

  • More parameters there are, more rich the problem is;

  • Notations :

    • \(e^i(\mu)\) is the relative error on the output when \(i\) parameters vary.

image::Relative-error-2.png